In the documentation here, it appears as though there is a “System Administrator” permission level required to use Tulip’s Table API feature. However, our Account Admin user does not see a permission level with that name. He’s promoted me to “Administrator” but I still do not have access to the “Settings” button described in the Table API documentation. Here is what I see vs what the documentation says I should see:
hello @evan.mccann, and welcome to the Tulip Community!!
great question, the site Settings are only accessible to Account Owners. for more information on what assets each User has access to, reference the following Support article: https://support.tulip.co/en/articles/2259988-managing-user-roles-on-the-people-page 
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can you ask the Account Owner to assign your User role to an Account Owner (there can be more than 1 per Tulip Account)?? if your Tulip account does not have an Account Owner, feel free to reach out to me via direct message and we can discuss there.
and again, welcome to the Tulip Community!!
gio